314 XI ELEMENTARY SPECTRAL THEORY

(11.1.2) Suppose E is a complex Banach space, u a continuous operator in
E. The set Ru of regular elements C e C for u is open in C and the mapping
£ _> (u - f . IE)-I o/Ru /flfo jjf(E) ft analytic.

Suppose Co e Rw» and let ^o = (u -Co * IE)"*- For anY C e C, we may
write, in J?(E),

H - C • 1E = H - Co ' IB - (C - Co) ' IE = (H - Co ' IE)(! - (C - Co)»o)-
But, by (83.2.1), for |C - Col < Kll~', IE - (C - CoK has an inverse in & (E),

00

equal to the sum of the absolute convergent series £ (C — £0)^0 > hence, for

w = 0

these values of C, w - C • IE is invertible in JSf(E), and its inverse, equal to
(1E - (C ~ Co>o)~X, can be written (u - C • IE)"' = f (C - Co)w»o + 1. the

71=0

series being absolutely convergent for |( — Col < ll^oll"1 '•> which ends the
proof.

(11.1.3) IfE is a complex Banach space, the spectrum of any continuous
operator u in E is a nonempty compact subset of C contained in the ball

ICI < HI-

First observe that for C ^ 0, u - C * 1E = ~C(1E ~ C"1"), and therefore
M - C • IE is invertible in J?(E) for |C| > ||w||, by (8.3.2.1). Furthermore, for

ICI > IMI,

where the series is absolutely convergent, and

as soon as |(| ^ 2||M||, we have therefore \\(u — C • 1E)~"1 \\ < \\u\\ ~1. Now, if we
had Ru = C, (u — C * Is)"1 would be an entire function (9.9.6), bounded in C
since it is bounded in the compact set |(| < 2||w|| and bounded in its comple-
ment ; by Li ouville's theorem (9.11.1)(w — C • IE)~* would be a constant, hence
also its inverse u — C • 1E , which is absurd. The first part of the proof shows
in addition that (u — C * IE)^ exists and is analytic for |(| > ||w||, therefore
the spectrum of w, which is closed in C, is compact and contained in the ball

ICI < [Ml-

It is possible to give examples of operators for which the spectrum is an
arbitrary compact subset of C (see Problem 3). not